This information can be found to much detail in ICAO Annex 10.
DME transmissions are sent in pairs of two pulses, in order to make it possible to use DMEs with the same frequency closer together, as well as to minimize errors from noise or other aircraft DMEs. The time difference between pulses vary with different DME stations, see 3.5.4.4 in Annex 10.
Before the DME ground station (the transponder) replies, it pauses either 50, 56 or 62 microseconds, depending on the type of DME (the aircraft receiver will "know" this, you can see the letter next to the channel indicating either X, Y, W or Z). This delay can be adjusted to calibrate the DME, as well as adjust the distance readout for ILS or MLS.
Do not confuse the transmission delay with the delay between the pulses in the pulse pair.
Since it takes 12.36 microseconds for radio waves to travel one nautical mile and back (a "radar mile"), the 50 to 56 microseconds could cater for adjustments of several nautical miles, if desirable. If the aircraft is closer than this distance, I suppose the DME would read zero, but I have never tried it nor found anything about it. Usually the corrections are only a few tenths of a nautical mile, to make the DME read zero at the threshold. If the adjustment is more than a certain value it should be published for the DME.
How much does a few tenths of a mile matter to updating INS/IRS? Even if on a RNP1 airway, it shouldn't be a problem.
From Annex 10, page 150 or so:
7.1.6 Siting of DME associated with ILS or MLS
7.1.6.1 The DME should, where possible, provide to the pilot an indicated zero range at touchdown in order to satisfy current operational requirements.
7.1.6.2 The optimum site for a DME transponder is dependent upon a number of technical and operational factors. DME/N may be installed with ILS or MLS where operational requirements permit. DME/P, which provides higher accuracy and coverage throughout the entire runway region, is required to support the more flexible and advanced operations that are available with MLS.
7.1.6.3 In the case of DME/N, the provision of zero range indication may be achieved by siting the transponder as close as possible to the point at which zero range indication is required. Alternatively, the transponder time delay can be adjusted to permit aircraft interrogators to indicate zero range at a specified distance from the DME antenna. When the indicated DME zero range has a reference other than the DME antenna, consideration should be given to publishing this information.